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Question: How do you find the derivative of \(\dfrac{1}{{{x^2}}}\) ?...

How do you find the derivative of 1x2\dfrac{1}{{{x^2}}} ?

Explanation

Solution

Hint : First, we shall analyze the given information so that we can able to solve the problem. Generally in Mathematics, the derivative refers to the rate of change of a function with respect to a variable. Here, we are applying the power rule to find the required answer.
We often use the power rule to calculate the derivative of a variable raised to a power and the power rule is the most commonly used derivative rule.

Formula used:
The formula that is applied in the power rule is as follows.
ddx(xn)=nxn−1\dfrac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}

Complete step-by-step solution:
Here we are asked to find the derivative of 1x2\dfrac{1}{{{x^2}}}
To find: ddx1x2\dfrac{d}{{dx}}\dfrac{1}{{{x^2}}}
Now, we shall write 1x2\dfrac{1}{{{x^2}}} as x−2{x^{ - 2}}
Since the variable xx is raised to a power −2 - 2 , we can use the power rule.
We often use the power rule to calculate the derivative of a variable raised to a power and the power rule is the most commonly used derivative rule.
The formula that is applied in the power rule is as follows.
ddx(xn)=nxn−1\dfrac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}
Henceddx1x2=ddxx−2\dfrac{d}{{dx}}\dfrac{1}{{{x^2}}} = \dfrac{d}{{dx}}{x^{ - 2}}
Now, we shall apply the power rule.
ddxx−2=−2x−2−1\dfrac{d}{{dx}}{x^{ - 2}} = - 2{x^{ - 2 - 1}}
⇒ddxx−2=−2x−3\Rightarrow \dfrac{d}{{dx}}{x^{ - 2}} = - 2{x^{ - 3}}
⇒ddxx−2=−21x3\Rightarrow \dfrac{d}{{dx}}{x^{ - 2}} = - 2\dfrac{1}{{{x^{^3}}}}
⇒ddxx−2=−2x3\Rightarrow \dfrac{d}{{dx}}{x^{ - 2}} = \dfrac{{ - 2}}{{{x^{^3}}}}
⇒ddx1x2=−2x3\Rightarrow \dfrac{d}{{dx}}\dfrac{1}{{{x^2}}} = \dfrac{{ - 2}}{{{x^{^3}}}}
Hence the derivative of 1x2\dfrac{1}{{{x^2}}} =−2x3 = \dfrac{{ - 2}}{{{x^{^3}}}} which is the required answer.

Note: The power rule is one of the derivative rules such that if xx is a variable which is raised to a power nn , then the derivative of xx raised to the power is denoted by the formula, ddx(xn)=nxn−1\dfrac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}} . Also, we often use the power rule to calculate the derivative of a variable raised to a power and the power rule is the most commonly used derivative rule.